This paper proposes an elegant optimization framework consisting of a mix of linear-matrix-inequality and second-order-cone constraints. The proposed framework generalizes the semidefinite relaxation (SDR) enabled solution to the typical transmit beamforming problems presented in the form of quadratically constrained quadratic programs (QCQPs) in the literature. It is proved that the optimization problems subsumed under the framework always admit a rank-one optimal solution when they are feasible and their optimal solutions are not trivial. This finding indicates that the relaxation is tight as the optimal solution of the original beamforming QCQP can be straightforwardly obtained from that of the SDR counterpart without any loss of optimality. Four representative examples of transmit beamforming, i.e., transmit beamforming with perfect channel state information (CSI), transmit beamforming with imperfect CSI, chance-constraint approach for imperfect CSI, and reconfigurable-intelligent-surface (RIS) aided beamforming, are shown to demonstrate how the proposed optimization framework can be realized in deriving the SDR counterparts for different beamforming designs.

翻译：本文提出了一种优雅的优化框架，该框架由线性矩阵不等式与二阶锥约束混合构成。该框架推广了传统的基于半定松弛（SDR）的发射波束成形问题解决方案，这些波束成形问题在文献中通常以二次约束二次规划（QCQPs）形式呈现。文中证明了：当该框架下的优化问题可行且其最优解非平凡时，这些问题始终存在一个秩一的最优解。这一发现表明松弛是紧致的，因为原始波束成形QCQP问题的最优解可直接从对应的SDR问题的最优解获得，且不会损失任何最优性。通过四个代表性发射波束成形实例（即完美信道状态信息（CSI）下的发射波束成形、非完美CSI下的发射波束成形、针对非完美CSI的机会约束方法，以及可重构智能表面（RIS）辅助波束成形），展示了如何将该优化框架应用于推导不同波束成形设计的SDR对应问题。